Mark E. Limes

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Photon Shot Noise
Cramér Rao Decaying Sine Wave

Monetization- Hey, buy a friggin' MUG from my wife Heidi Cups

Photon Shot Noise

DC light incident on a photodiode creates a photocurrent that creates a voltage drop across an attached resistor.

Photon shot noise = \(\sqrt{2 e V R} \), where \(e = 1.60217\times 10^{-19}\) C is the electron charge, \(V\) is voltage measured, and \(R\) is the feedback resistor. These values also auto-seed the CRLB calculator below.

Voltage \(V\) = Volts

Resistance \(R_f\) = Ohms

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PSN = V/√ Hz
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The power \(P\) of the light incident on the photodiode is obtained from the current \(I \) divided by the efficiency of the photodiode \(\eta\).

\(P = I/\eta = \frac{V }{R \eta} \)

Photodiode efficiency \(\eta \) = A/W = Amperes/Watt

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\(P\) = W
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Cramér Rao Decaying Sine Wave

Cramér Rao Wikipedia entry

The best-case frequency error from fitting a monoexponetially decaying sine wave, \(y(t) = A \exp(-t/T_2) \sin(2\pi t + \phi) \), can be found using the Cramér Rao Lower Bound of the variance of unbiased estimators (minimum variance unbiased MVU estimator). The lower bound of the standard deviation of frequency is given by $$\sigma_f = \frac{ \sqrt{12 C}}{2\pi (A/\rho) T^{3/2}} \text{Hz , }$$ where \(\rho\) is the spectral noise density (V/√ Hz) of the measurement, \(A\) is the amplitude (V) of the sine wave, and \(T\) is the measurement time (seconds). The dimensionless constant \(C\) takes into account the exponential decay. Note \(A/\rho = \text{SNR}\) is the signal-to-noise ratio. For a non-decaying sine wave, \(C = 1\). Courtesy of Eq. 3 in V.G. Lucivero, et al., we have in the limit of a fast sampling time (or large number of samples per measurement), $$C = \frac{2 T^3 e^{2\alpha}(e^{2\alpha} - 1)}{3T_2(T_2^2e^{2\alpha}( e^{2\alpha} - 2) + T_2^2 -4 T^2 e^{2\alpha} ) }, \text{with } \alpha = T/T_2 . $$
Amplitude \(A\) = Volts

Spectral Noise Density \(\rho\) = V/√ Hz

Detection Time \(T\) = Seconds

Relaxation Time \(T_2\) = Seconds


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\(C\) =
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\(\sigma_f\) = Hz
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Often this frequency measurement is repeated with a repetition time of \(T_r\), giving a bandwidth of \(\text{BW}=1/2 T_r\). Thus, assuming white noise, an estimate of the frequency spectral density is \(\sigma_f /\sqrt{\text{BW}}\). For \(T_r = T\), this estimate becomes $$ \text{Spectral density} = \frac{ \sqrt{24 C} }{2\pi (A/\rho) T} \frac{ \text{Hz}}{\sqrt{\text{Hz}}}. $$ ------------------------------------------------------------
\(\sigma_f\)/√ BW = Hz/√ Hz
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Assume the frequency measurement is due to observing the Larmor precession of a quantum mechanical spin, with field coupling \(\gamma \mathbf{B}\cdot\mathbf{S}\). Here the coupling strength \(\gamma\) is the gyromagnetic ratio in Hz/Tesla (see table below), (Gyromagnetic ratio Wikipedia entry.) \(\mathbf{B}\) is the field strength in Tesla, and \(\mathbf{S}\) is the spin state. The estimate of a magnetometer spectral density for a system using this spin species is then $$ \text{Mag. Spectral density} = \frac{ \sqrt{24 C} }{2\pi \gamma (A/\rho) T} \frac{ \text{T}}{\sqrt{\text{Hz}}}. $$

Gyromagnetic Ratio \(\gamma\) = Hz/T

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\(\sigma_B/√ BW \) = T/√Hz,
in a bandwidth of BW = Hz
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Gyromagnetic ratio list
--------Isotope-------- --------Style------- --------\(\gamma\) [Hz/T]--------
Rubidium 87, \(^{87}\)Rb Electronic-Nuclear coupled, low field   ~7e9
Rubidium 85, \(^{85}\)Rb Electronic-Nuclear coupled, low field   ~4.666e9
Bare Electron, e Electron   28.02495164e9
Hydrogen, H\(_{2}\)O Nuclear   42.57638474e6
Helium 3, \(^{3}\)He Nuclear   -32.43409942e6
Xenon 129, \(^{129}\)Xe Nuclear   -11.777e6


Interestingly, the energy resolution lower bound estimate per measurement is
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\(\sigma_E\) = eV
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